# Partial Correlation

Suspected correlation is a problem to analyze correlation.

Partial-correlation-coefficient (PCC) is a kind of correlation-coefficient. We study the effect of suspected correlation.

## Calculation of Partial Correlation Coefficient

This example is that there are variables X, Y, S and T. We want to study the relationship between X and Y.

At first, we calculate X' by Multi-Regression Analysis .
X' = A * S + B * T + C
X' is the predicted value.

For example, there is the data set, (X1, Y1, S1, T1)
X1' = A * S1 + B * T1 + C
is calculated.

X1 = X1' + Ex1
Ex means a residual.

Ex1 = X1 - X1'
Ex is modified X. The effect of S and T is deleted.

Ey is calculated in the same way.

PCC is the coefficient of Ex and Ey. The coefficient is calculated by the data set of (Ex1, Ey1), (Ex2, Ey2), (Ex3, Ey3), (Ex4, Ey4) and so on.

PCC is a general definition of the correlation coefficient.

## From Partial Correlation Coefficient Matrix to Graphical Modeling

The correlation coefficient matrix is easy to make and useful. The analysis of the matrix has weak point.

The PCC-matrix covers the weak point. But the calculation of the matrix is not easy.

Selection of variance is important not only in the multi-regression analysis but the analysis of partial correlation coefficient. Graphical modeling is one of the applications of such analysis.

## Difficulty of Partial-Correlation-Coefficient

### Difficulty of Calculation

PCC is useful but the calculation is not easy.

And PCC includes the problem of Selection of Variance and Multicollinearity because it uses multi regression analysis.

### Weakness of Partial-Correlation-Coefficient-Matrix

General calculation of PCC-matrix uses correlation matrix. And it is not calculated if data set includes Multicollinearity .

We want to analyze the suspected correlation in the strong correlation pair by PCC-matrix. But PCC-matrix cannot analyze such strong relationship. This is the largest weakness of PCC-matrix.

When we want to analyze the suspected correlation in the strong correlation pair by PCC, we calculate only the PCC of the pair by multi regression analysis politely.

### Difficulty of Suspected Correlation

The analysis of Suspected Correlation is not done without the knowledge of physical meaning of the variables.

We need statistical approach and physical approach like the wheel of the car.

When we have a little knowledge of physical relationship, we can add some idea in the sampling of variables.